Hoje vou à clínica para fazer um exame de rotina.

In Brazilian Portuguese, the present tense of ir + infinitive is the most common way to express a near or planned future.

• Hoje vou... = Today I’m going to...
  It often sounds more natural than irei (simple future), which can sound more formal or less common in everyday speech.

à is a contraction of a (the preposition to) + a (the feminine singular definite article the).

• vou a + a clínica → vou à clínica
  You use this because ir commonly takes the preposition a in this meaning (go to), and clínica is feminine, so it takes a as the article (a clínica = the clinic).

A quick test: if you can replace the destination noun with a masculine one and it becomes ao (a + o), then the feminine version would be à.

• Vou ao médico (to the doctor, masculine)
• Vou à clínica (to the clinic, feminine)
  If there’s no article involved, you may only need a (no accent), but with many common places Portuguese does use the article.

Clínica is a general word for a clinic or medical facility, often for appointments and exams. It can be public or private depending on context.
Related options you may see:

• consultório = a doctor’s office (often one practitioner’s office)
• hospital = hospital
  So clínica works well when you’re going for tests like um exame.

para means in order to / to (purpose).

• para fazer um exame = to do/perform a test (the reason you’re going)
  Without para, it would sound incomplete or change structure; Portuguese typically marks purpose with para + infinitive.

Yes. In everyday Brazilian Portuguese, para often becomes pra (informal speech and casual writing).

• Hoje vou à clínica pra fazer um exame de rotina.
  For formal writing, stick with para.

exame de rotina means a routine exam/check-up, i.e., not because of a specific emergency or symptom, but as a regular health check.
You may also hear:

• check-up (borrowed word, common in Brazil)
• exames de rotina (plural, common if multiple tests are involved)

um exame introduces the exam as “a routine exam” (not a specific one already known to the listener).
You might use o exame if both speakers already know which exam it is:

• Vou fazer o exame que o médico pediu. (the exam the doctor ordered)

Yes, it’s flexible, with small emphasis changes:

• Hoje vou à clínica... (emphasis on today)
• Vou à clínica hoje... (still natural; today feels like an added time detail)
• Eu vou hoje à clínica... (possible, but a bit more marked/less common in casual speech)